large inclusions for groupoid and k-graph c*-algebras -...

Large inclusions for groupoid and k-graph

C ∗-algebrasPreliminary report

Danny Crytser

St. Lawrence University

JMM 2019

Danny Crytser ( St. Lawrence University) Large inclusions for groupoid and k-graph C∗-algebras JMM 2019 1 / 12

What is a “large” inclusion

Let B ⊂ A be a unital C ∗-subalgebra. (1A ∈ B)

regular: span{n ∈ A : nBn∗ ∪ n∗Bn ⊂ B} = Aat most one conditional expectation P : A→ B (ucp mapequaling the identity on B);

unique pseudo-expectation ψ : A→ I (B), where I (B) is theinjective envelope of B

Often motivated by simplicity criteria and ideal structure problems(see [Zar]).



Let B ⊂ A be a unital C ∗-subalgebra. (1A ∈ B)regular: span{n ∈ A : nBn∗ ∪ n∗Bn ⊂ B} = A

at most one conditional expectation P : A→ B (ucp mapequaling the identity on B);





Let B ⊂ A be a unital C ∗-subalgebra. (1A ∈ B)regular: span{n ∈ A : nBn∗ ∪ n∗Bn ⊂ B} = Aat most one conditional expectation P : A→ B (ucp mapequaling the identity on B);




What is a pseudo-expectation?

Definition

Let A be a C ∗-algebra. Then an injective envelope for A consistsof an injective C ∗-algebra I (A) containing A as a C ∗-subalgebra theonly ucp map I (A)→ I (A) that restricts to the identity on A is theidentity map. (It exists and is unique up to isomorphism ([Ham79]).)

Definition ([Pit12])

A pseudo-expectation for B ⊂ A is a ucp map A→ I (B) extendingB ↪→ I (B). (Generalizes conditional expectation, always exists.)

Shown in [Pit12] that regular MASA inclusions have uniquepseudo-expectations.


What is a groupoid

Groupoid = “category where every morphism is invertible”. So ifα, β ∈ G then αβ may or may not be defined. Can always cancel,e.g. α−1(αβ) = β.

Important subsets:unit space G (0): set of elements u such that u2 = u (equal torange of α 7→ α−1α)isotropy bundle: elements α such that α−1α = αα−1

Can add topology to make everything continuous. If α 7→ αα−1 is alocal homeomorphism G → G , call G étale. Care especially aboutthe interior of isotropy bundle: Int IsoG . (If G is étale thisincludes the unit space.)

Example

If discrete group G y X a topological space, then G × X is agroupoid with (h, g .x)(g , x) = (hg , x). (Call this G n X ,transformation groupoid.)


What is a groupoid

Groupoid = “category where every morphism is invertible”. So ifα, β ∈ G then αβ may or may not be defined. Can always cancel,e.g. α−1(αβ) = β. Important subsets:

unit space G (0): set of elements u such that u2 = u (equal torange of α 7→ α−1α)

isotropy bundle: elements α such that α−1α = αα−1


Example



What is a groupoid

Groupoid = “category where every morphism is invertible”. So ifα, β ∈ G then αβ may or may not be defined. Can always cancel,e.g. α−1(αβ) = β. Important subsets:

unit space G (0): set of elements u such that u2 = u (equal torange of α 7→ α−1α)isotropy bundle: elements α such that α−1α = αα−1


Example



Groupoid C ∗-algebras

Let G be an étale locally compact Hausdorff second countablegroupoid with compact unit space. Construct reduced groupoidC ∗-algebra C ∗r (G ) out of convolution algebra Cc(G ) as inconstruction of group C ∗-algebra.

Important inclusions related to this

C (G (0)) ⊂ C ∗r (G ) C ∗r (Int IsoG ) ⊂ C ∗r (G )

Example (Graph algebras)

If E is a graph with path groupoid GE , then these correspond to thediagonal C ∗(sλs

∗λ) ⊂ C ∗(E ) and the abelian core ME ⊂ C ∗(E )

(Nagy-Reznikoff).


Groupoid C ∗-algebras

Let G be an étale locally compact Hausdorff second countablegroupoid with compact unit space. Construct reduced groupoidC ∗-algebra C ∗r (G ) out of convolution algebra Cc(G ) as inconstruction of group C ∗-algebra. Important inclusions related to this

C (G (0)) ⊂ C ∗r (G ) C ∗r (Int IsoG ) ⊂ C ∗r (G )

Example (Graph algebras)

If E is a graph with path groupoid GE , then these correspond to thediagonal C ∗(sλs

∗λ) ⊂ C ∗(E ) and the abelian core ME ⊂ C ∗(E )

(Nagy-Reznikoff).


Main questions for this work in progress:

Question

When do the inclusions C (G (0)) ⊂ C ∗r (G ) andC ∗r (Int IsoG ) ⊂ C ∗r (G ) associated to an étale groupoid G have thevarious largeness properties on the first slide?.

Specifically, when does C ∗r (Int IsoG ) ⊂ C ∗r (G ) have the faithfulunique pseudo-expectation property?


The inclusion C (G (0)) ⊂ C ∗r (G )

Proposition (C.)

The inclusion C (G (0)) ⊂ C ∗r (G ). Following are equivalent:(a) has a unique conditional expectation

(b) has a unique pseudo-expectation

(c) Int IsoG = G (0)

Proof.

Easy proof, combining results of Renault, Pitts, and Zarikian. Avoidsconsidering properly outer actions as in Zarikian.


The inclusion C ∗r (Int IsoG ) ⊂ C ∗r (G )

The inclusion C ∗r (Int IsoG ) ⊂ C ∗r (G ) is always regular (noted bymany, including [CN]).

Proposition ([BNR+16, Prop. 4.1])

The inclusion C ∗r (Int IsoG ) ⊂ C ∗r (G ) has a conditional expectation ifand only if Int IsoG is closed in G . The expectation is given onCc(G ) by f 7→ f |Int IsoG .

This conditional expectation is unique using an argument of [Zar18]

Question

What about pseudo-expectations?

The inclusion C ∗r (Int IsoG ) ⊂ C ∗r (G ) is essential ([BNR+16]), leadsus to hope for unique faithful pseudo-expectation.


Maximal abelian subalgebras and

pseudoexpectations

Theorem ([BNR+16, Thm. 4.3])

Suppose that Int IsoG is abelian. If Int IsoG is closed or if thereexists a countable discrete abelian group H and a continuous1-cocycle c : G → H that is injective on each G xx , thenC ∗r (Int IsoG ) ⊂ C ∗r (G ) is a MASA

Any essential MASA has unique faithful pseudo-expectation [Pit12],we obtain.

Proposition (C.)

In either of the above cases, there is a unique pseudo-expectationfrom C ∗r (G )→ I (C ∗r (Int IsoG )), which is faithful

Hope that the above is always true, not just for MASA C ∗r (Int IsoG ).Danny Crytser ( St. Lawrence University) Large inclusions for groupoid and k-graph C∗-algebras JMM 2019 9 / 12

The End

Thank You!


Bibliography I

J. Brown, G. Nagy, S. Reznikoff, A. Sims, and D. Williams.Cartan subalgebras in C ∗-algebras of Hausdorff étale Groupoids.Integral Equations and Operator Theory, 85:109–126, 2016.

D. Crytser and G. Nagy.Simplicity criteria for groupoid C ∗-algebras.J. Operator Theory.to appear.

M. Hamana.Injective envelopes of C ∗-algebras.J. Math. Soc. Japan, 31(1):181–197, 1979.


Bibliography II

D. Pitts.Structure for regular inclusions.J. Operator Theory, 78(2), 2012.

V. Zarikian.Unique expectations for discrete crossed products.Annals Func. Anal., to appear.

V. Zarikian.Unique extension problems for C ∗-inclusions.GPOTS, 2018.


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